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Properties

Label 12.0.14560594398949509.2
Degree $12$
Signature $[0, 6]$
Discriminant $3^{7}\cdot 367^{5}$
Root discriminant $22.23$
Ramified primes $3, 367$
Class number $1$
Class group Trivial
Galois group $\GL(2,Z/4)$ (as 12T52)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![171, 297, -108, -306, 4, 99, 80, -60, -24, 15, 8, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 6*x^11 + 8*x^10 + 15*x^9 - 24*x^8 - 60*x^7 + 80*x^6 + 99*x^5 + 4*x^4 - 306*x^3 - 108*x^2 + 297*x + 171)
 
gp: K = bnfinit(x^12 - 6*x^11 + 8*x^10 + 15*x^9 - 24*x^8 - 60*x^7 + 80*x^6 + 99*x^5 + 4*x^4 - 306*x^3 - 108*x^2 + 297*x + 171, 1)
 

Normalized defining polynomial

\( x^{12} - 6 x^{11} + 8 x^{10} + 15 x^{9} - 24 x^{8} - 60 x^{7} + 80 x^{6} + 99 x^{5} + 4 x^{4} - 306 x^{3} - 108 x^{2} + 297 x + 171 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $12$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(14560594398949509=3^{7}\cdot 367^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.23$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $3, 367$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{12} a^{8} + \frac{1}{6} a^{7} - \frac{1}{12} a^{6} + \frac{5}{12} a^{5} - \frac{1}{3} a^{4} + \frac{5}{12} a^{3} + \frac{1}{12} a^{2} - \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{12} a^{9} + \frac{1}{12} a^{7} + \frac{1}{12} a^{6} + \frac{1}{3} a^{5} - \frac{5}{12} a^{4} + \frac{1}{4} a^{3} - \frac{5}{12} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{2208} a^{10} - \frac{5}{2208} a^{9} - \frac{1}{368} a^{8} + \frac{9}{368} a^{7} + \frac{7}{184} a^{6} - \frac{77}{368} a^{5} + \frac{535}{1104} a^{4} + \frac{911}{2208} a^{3} + \frac{39}{736} a^{2} + \frac{37}{184} a - \frac{239}{736}$, $\frac{1}{68448} a^{11} + \frac{5}{34224} a^{10} + \frac{893}{22816} a^{9} + \frac{451}{17112} a^{8} - \frac{7373}{34224} a^{7} - \frac{7421}{34224} a^{6} - \frac{6617}{17112} a^{5} + \frac{7945}{68448} a^{4} - \frac{4793}{34224} a^{3} + \frac{23359}{68448} a^{2} + \frac{9525}{22816} a + \frac{10767}{22816}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $5$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{55}{2208} a^{10} + \frac{275}{2208} a^{9} - \frac{37}{368} a^{8} - \frac{127}{368} a^{7} + \frac{29}{184} a^{6} + \frac{463}{368} a^{5} - \frac{721}{1104} a^{4} - \frac{3185}{2208} a^{3} - \frac{1593}{736} a^{2} + \frac{587}{184} a + \frac{2657}{736} \) (order $6$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 13723.3534743 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$\GL(2,Z/4)$ (as 12T52):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 96
The 14 conjugacy class representatives for $\GL(2,Z/4)$
Character table for $\GL(2,Z/4)$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.3.1101.1, 6.0.3636603.4

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 12 siblings: 12.6.5343738144414469803.1, 12.0.14560594398949509.3, 12.0.14560594398949509.4
Degree 16 siblings: Deg 16, 16.0.1298528369092716162129.1
Degree 24 siblings: data not computed
Degree 32 sibling: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.4.0.1}{4} }{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
367Data not computed